Linear.Quaternion:$ctan from linear-1.19.1.3

Percentage Accurate: 84.5% → 99.8%

Time: 16.7s

Alternatives: 18

Speedup: 2.3×

• 64.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \frac{y}{x}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (cosh(x) * (y / x)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (Math.cosh(x) * (y / x)) / z;
}

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (cosh(x) * (y / x)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Initial Program: 84.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \frac{y}{x}}{z} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (* (cosh x) (/ y x)) z))

C

double code(double x, double y, double z) {
	return (cosh(x) * (y / x)) / z;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (cosh(x) * (y / x)) / z
end function

Java

public static double code(double x, double y, double z) {
	return (Math.cosh(x) * (y / x)) / z;
}

Python

def code(x, y, z):
	return (math.cosh(x) * (y / x)) / z

Julia

function code(x, y, z)
	return Float64(Float64(cosh(x) * Float64(y / x)) / z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = (cosh(x) * (y / x)) / z;
end

Wolfram

code[x_, y_, z_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;z\_m \leq 1.12 \cdot 10^{+24}:\\ \;\;\;\;\frac{\cosh x \cdot \frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= z_m 1.12e+24)
    (/ (* (cosh x) (/ y z_m)) x)
    (* y (/ (/ (cosh x) x) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.12e+24) {
		tmp = (cosh(x) * (y / z_m)) / x;
	} else {
		tmp = y * ((cosh(x) / x) / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (z_m <= 1.12d+24) then
        tmp = (cosh(x) * (y / z_m)) / x
    else
        tmp = y * ((cosh(x) / x) / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (z_m <= 1.12e+24) {
		tmp = (Math.cosh(x) * (y / z_m)) / x;
	} else {
		tmp = y * ((Math.cosh(x) / x) / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if z_m <= 1.12e+24:
		tmp = (math.cosh(x) * (y / z_m)) / x
	else:
		tmp = y * ((math.cosh(x) / x) / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (z_m <= 1.12e+24)
		tmp = Float64(Float64(cosh(x) * Float64(y / z_m)) / x);
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / x) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (z_m <= 1.12e+24)
		tmp = (cosh(x) * (y / z_m)) / x;
	else
		tmp = y * ((cosh(x) / x) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[z$95$m, 1.12e+24], N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / z$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1.12e24

   1. Initial program 83.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. div-inv N/A

         \[\leadsto \color{blue}{\left(\cosh x \cdot \frac{y}{x}\right) \cdot \frac{1}{z}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{x}} \cdot \frac{1}{z} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\cosh x \cdot y\right) \cdot \frac{1}{z}}{x}} \]

      5. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \left(y \cdot \frac{1}{z}\right)}}{x} \]

      6. div-inv N/A

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{z}}}{x} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x \cdot \frac{y}{z}}}{x} \]

      8. cosh-lowering-cosh.f64 N/A

         \[\leadsto \frac{\color{blue}{\cosh x} \cdot \frac{y}{z}}{x} \]

      9. /-lowering-/.f64 93.5

         \[\leadsto \frac{\cosh x \cdot \color{blue}{\frac{y}{z}}}{x} \]

   4. Applied egg-rr 93.5%

      \[\leadsto \color{blue}{\frac{\cosh x \cdot \frac{y}{z}}{x}} \]

   if 1.12e24 < z

   1. Initial program 87.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

      2. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \cosh x}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} \cdot \cosh x\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x \cdot \frac{1}{x}}}{z} \]

      8. div-inv N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      10. cosh-lowering-cosh.f64 99.8

          \[\leadsto y \cdot \frac{\frac{\color{blue}{\cosh x}}{x}}{z} \]

   4. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{x}}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 93.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 2 \cdot 10^{+203}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\cosh x}{x}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (* (cosh x) (/ y x)) 2e+203)
    (*
     (/ y x)
     (/
      (fma (fma x x 0.0) (fma x (fma x 0.041666666666666664 0.0) 0.5) 1.0)
      z_m))
    (* y (/ (/ (cosh x) x) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((cosh(x) * (y / x)) <= 2e+203) {
		tmp = (y / x) * (fma(fma(x, x, 0.0), fma(x, fma(x, 0.041666666666666664, 0.0), 0.5), 1.0) / z_m);
	} else {
		tmp = y * ((cosh(x) / x) / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y / x)) <= 2e+203)
		tmp = Float64(Float64(y / x) * Float64(fma(fma(x, x, 0.0), fma(x, fma(x, 0.041666666666666664, 0.0), 0.5), 1.0) / z_m));
	else
		tmp = Float64(y * Float64(Float64(cosh(x) / x) / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], 2e+203], N[(N[(y / x), $MachinePrecision] * N[(N[(N[(x * x + 0.0), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664 + 0.0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[Cosh[x], $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 2e203

   1. Initial program 94.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 88.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 88.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right)}}{z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1}{z}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1}{z}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1}{z} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{y}{x} \cdot \color{blue}{\frac{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1}{z}} \]

   7. Applied egg-rr 89.3%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)}{z}} \]

   if 2e203 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 65.1%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

      2. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \cosh x}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} \cdot \cosh x\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x \cdot \frac{1}{x}}}{z} \]

      8. div-inv N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      10. cosh-lowering-cosh.f64 99.9

          \[\leadsto y \cdot \frac{\frac{\color{blue}{\cosh x}}{x}}{z} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{x}}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z\_m} \leq 10^{-98}:\\ \;\;\;\;\frac{y}{z\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* (cosh x) (/ y x)) z_m) 1e-98)
    (/ y (* z_m x))
    (/
     (/
      (*
       y
       (fma
        (* x x)
        (fma
         (* x x)
         (fma (* x x) 0.001388888888888889 0.041666666666666664)
         0.5)
        1.0))
      z_m)
     x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (((cosh(x) * (y / x)) / z_m) <= 1e-98) {
		tmp = y / (z_m * x);
	} else {
		tmp = ((y * fma((x * x), fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)) / z_m) / x;
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y / x)) / z_m) <= 1e-98)
		tmp = Float64(y / Float64(z_m * x));
	else
		tmp = Float64(Float64(Float64(y * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)) / z_m) / x);
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 1e-98], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 9.99999999999999939e-99

   1. Initial program 93.4%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 58.7

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 58.7%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. *-lowering-*.f64 58.7

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

   7. Applied egg-rr 58.7%

      \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

   if 9.99999999999999939e-99 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 73.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

      2. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \cosh x}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} \cdot \cosh x\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x \cdot \frac{1}{x}}}{z} \]

      8. div-inv N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      10. cosh-lowering-cosh.f64 93.8

          \[\leadsto y \cdot \frac{\frac{\color{blue}{\cosh x}}{x}}{z} \]

   4. Applied egg-rr 93.8%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{x}}{z}} \]

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}}{z} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}}{z} \]

   7. Simplified 89.0%

      \[\leadsto y \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}}{z} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1}{x \cdot z}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1\right)}{x \cdot z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1\right)}{\color{blue}{z \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1\right)}{z}}{x}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1\right)}{z}}{x}} \]

   9. Applied egg-rr 95.9%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0\right), 0.5\right), 1\right) \cdot y}{z}}{x}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot y}{z}}{x} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot y}{z}}{x} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \cdot y}{z}}{x} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot y}{z}}{x} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot y}{z}}{x} \]

       5. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \cdot y}{z}}{x} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot y}{z}}{x} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       9. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       10. *-commutative N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       11. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       12. unpow2 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       13. *-lowering-*.f64 95.9

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot y}{z}}{x} \]

   12. Simplified 95.9%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot y}{z}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 10^{-98}:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z\_m} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (/ (* (cosh x) (/ y x)) z_m) 2e+305)
    (/
     (/ (* y (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0)) x)
     z_m)
    (*
     y
     (/
      (/
       (fma
        (* x x)
        (fma
         x
         (* x (fma (* x x) 0.001388888888888889 0.041666666666666664))
         0.5)
        1.0)
       x)
      z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (((cosh(x) * (y / x)) / z_m) <= 2e+305) {
		tmp = ((y * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0)) / x) / z_m;
	} else {
		tmp = y * ((fma((x * x), fma(x, (x * fma((x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / x) / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(Float64(cosh(x) * Float64(y / x)) / z_m) <= 2e+305)
		tmp = Float64(Float64(Float64(y * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0)) / x) / z_m);
	else
		tmp = Float64(y * Float64(Float64(fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664)), 0.5), 1.0) / x) / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], 2e+305], N[(N[(N[(y * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z) < 1.9999999999999999e305

   1. Initial program 94.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 89.0

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 89.0%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot 1} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x}}{z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y} + \frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x}}{z} \]

      4. distribute-rgt-out N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{\left(y \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right)} \cdot {x}^{2}}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot {x}^{2}}{x}}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)}}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + y \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{x}}{z} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{x}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}{x}}}{z} \]

   8. Simplified 89.0%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right) \cdot y}{x}}}{z} \]

   if 1.9999999999999999e305 < (/.f64 (*.f64 (cosh.f64 x) (/.f64 y x)) z)

   1. Initial program 63.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

      2. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \cosh x}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} \cdot \cosh x\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x \cdot \frac{1}{x}}}{z} \]

      8. div-inv N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      10. cosh-lowering-cosh.f64 100.0

          \[\leadsto y \cdot \frac{\frac{\color{blue}{\cosh x}}{x}}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{x}}{z}} \]

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}}{z} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}}{z} \]

   7. Simplified 93.3%

      \[\leadsto y \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}}{z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\cosh x \cdot \frac{y}{x}}{z} \leq 2 \cdot 10^{+305}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 91.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;\cosh x \cdot \frac{y}{x} \leq 5 \cdot 10^{+204}:\\ \;\;\;\;\frac{y}{x} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right), 1\right)}{x}}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= (* (cosh x) (/ y x)) 5e+204)
    (*
     (/ y x)
     (/
      (fma (fma x x 0.0) (fma x (fma x 0.041666666666666664 0.0) 0.5) 1.0)
      z_m))
    (*
     y
     (/
      (/ (fma (* x x) (* x (* 0.001388888888888889 (* x (* x x)))) 1.0) x)
      z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if ((cosh(x) * (y / x)) <= 5e+204) {
		tmp = (y / x) * (fma(fma(x, x, 0.0), fma(x, fma(x, 0.041666666666666664, 0.0), 0.5), 1.0) / z_m);
	} else {
		tmp = y * ((fma((x * x), (x * (0.001388888888888889 * (x * (x * x)))), 1.0) / x) / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (Float64(cosh(x) * Float64(y / x)) <= 5e+204)
		tmp = Float64(Float64(y / x) * Float64(fma(fma(x, x, 0.0), fma(x, fma(x, 0.041666666666666664, 0.0), 0.5), 1.0) / z_m));
	else
		tmp = Float64(y * Float64(Float64(fma(Float64(x * x), Float64(x * Float64(0.001388888888888889 * Float64(x * Float64(x * x)))), 1.0) / x) / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[N[(N[Cosh[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], 5e+204], N[(N[(y / x), $MachinePrecision] * N[(N[(N[(x * x + 0.0), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664 + 0.0), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * N[(0.001388888888888889 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (cosh.f64 x) (/.f64 y x)) < 5.00000000000000008e204

   1. Initial program 94.6%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 88.8

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 88.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right)}}{z} \]

      2. associate-/l* N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1}{z}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1}{z}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x}} \cdot \frac{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1}{z} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \frac{y}{x} \cdot \color{blue}{\frac{\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1}{z}} \]

   7. Applied egg-rr 89.4%

      \[\leadsto \color{blue}{\frac{y}{x} \cdot \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)}{z}} \]

   if 5.00000000000000008e204 < (*.f64 (cosh.f64 x) (/.f64 y x))

   1. Initial program 64.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

      2. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \cosh x}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} \cdot \cosh x\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x \cdot \frac{1}{x}}}{z} \]

      8. div-inv N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      10. cosh-lowering-cosh.f64 99.9

          \[\leadsto y \cdot \frac{\frac{\color{blue}{\cosh x}}{x}}{z} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{x}}{z}} \]

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}}{z} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}}{z} \]

   7. Simplified 93.4%

      \[\leadsto y \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}}{z} \]

   8. Taylor expanded in x around inf

      \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{4}}, 1\right)}{x}}{z} \]
   9. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}, 1\right)}{x}}{z} \]

      2. pow-sqr N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \frac{1}{720} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}, 1\right)}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}}, 1\right)}{x}}{z} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2}\right)}, 1\right)}{x}}{z} \]

      5. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{720} \cdot {x}^{2}\right), 1\right)}{x}}{z} \]

      6. associate-*l* N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{720} \cdot {x}^{2}\right)\right)}, 1\right)}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{720} \cdot {x}^{2}\right) \cdot x\right)}, 1\right)}{x}}{z} \]

      8. associate-*r* N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot \left({x}^{2} \cdot x\right)\right)}, 1\right)}{x}}{z} \]

      9. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{720} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)\right), 1\right)}{x}}{z} \]

      10. unpow3 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{720} \cdot \color{blue}{{x}^{3}}\right), 1\right)}{x}}{z} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{720} \cdot {x}^{3}\right)}, 1\right)}{x}}{z} \]

      12. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\frac{1}{720} \cdot {x}^{3}\right)}, 1\right)}{x}}{z} \]

      13. cube-mult N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{720} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right), 1\right)}{x}}{z} \]

      14. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{720} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right), 1\right)}{x}}{z} \]

      15. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{720} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right), 1\right)}{x}}{z} \]

      16. unpow2 N/A

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(\frac{1}{720} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right), 1\right)}{x}}{z} \]

      17. *-lowering-*.f64 93.4

          \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, x \cdot \left(0.001388888888888889 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right), 1\right)}{x}}{z} \]

   10. Simplified 93.4%

       \[\leadsto y \cdot \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(0.001388888888888889 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}, 1\right)}{x}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 89.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00028:\\ \;\;\;\;\frac{\frac{y}{z\_m} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= x 0.00028)
    (/
     (*
      (/ y z_m)
      (fma x (* x (fma x (fma x 0.041666666666666664 0.0) 0.5)) 1.0))
     x)
    (if (<= x 1.25e+48)
      (* y (/ (cosh x) (* z_m x)))
      (/
       (/
        (*
         y
         (fma
          (* x x)
          (fma
           (* x x)
           (fma (* x x) 0.001388888888888889 0.041666666666666664)
           0.5)
          1.0))
        z_m)
       x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00028) {
		tmp = ((y / z_m) * fma(x, (x * fma(x, fma(x, 0.041666666666666664, 0.0), 0.5)), 1.0)) / x;
	} else if (x <= 1.25e+48) {
		tmp = y * (cosh(x) / (z_m * x));
	} else {
		tmp = ((y * fma((x * x), fma((x * x), fma((x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)) / z_m) / x;
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 0.00028)
		tmp = Float64(Float64(Float64(y / z_m) * fma(x, Float64(x * fma(x, fma(x, 0.041666666666666664, 0.0), 0.5)), 1.0)) / x);
	elseif (x <= 1.25e+48)
		tmp = Float64(y * Float64(cosh(x) / Float64(z_m * x)));
	else
		tmp = Float64(Float64(Float64(y * fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664), 0.5), 1.0)) / z_m) / x);
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 0.00028], N[(N[(N[(y / z$95$m), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + 0.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 1.25e+48], N[(y * N[(N[Cosh[x], $MachinePrecision] / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision] / x), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.7999999999999998e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 79.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 79.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* N/A

         \[\leadsto \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      3. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{\color{blue}{x \cdot z + 0}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x \cdot z + 0}} \]

   7. Applied egg-rr 77.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right) \cdot \frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{\color{blue}{x \cdot z}} \]

      2. associate-/l/ N/A

         \[\leadsto \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{z}}{x}} \]

   9. Applied egg-rr 86.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right) \cdot \frac{y}{z}}{x}} \]

   if 2.7999999999999998e-4 < x < 1.24999999999999993e48

   1. Initial program 100.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      2. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \cosh x}}{z \cdot x} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\cosh x}{z \cdot x}} \]

      7. cosh-lowering-cosh.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x}}{z \cdot x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

      9. *-lowering-*.f64 71.4

         \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   4. Applied egg-rr 71.4%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]

   if 1.24999999999999993e48 < x

   1. Initial program 75.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

      2. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \cosh x}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} \cdot \cosh x\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x \cdot \frac{1}{x}}}{z} \]

      8. div-inv N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      10. cosh-lowering-cosh.f64 100.0

          \[\leadsto y \cdot \frac{\frac{\color{blue}{\cosh x}}{x}}{z} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{x}}{z}} \]

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}}{z} \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}{x}}}{z} \]

   7. Simplified 98.2%

      \[\leadsto y \cdot \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{x}}}{z} \]
   8. Step-by-step derivation

      1. associate-/r* N/A

         \[\leadsto y \cdot \color{blue}{\frac{\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1}{x \cdot z}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1\right)}{x \cdot z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1\right)}{\color{blue}{z \cdot x}} \]

      4. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1\right)}{z}}{x}} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{720} + \frac{1}{24}\right)\right) + \frac{1}{2}\right) + 1\right)}{z}}{x}} \]

   9. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, \mathsf{fma}\left(x, x \cdot 0.001388888888888889, 0.041666666666666664\right), 0\right), 0.5\right), 1\right) \cdot y}{z}}{x}} \]

   10. Taylor expanded in x around 0

       \[\leadsto \frac{\frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} \cdot y}{z}}{x} \]
   11. Step-by-step derivation

       1. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1\right)} \cdot y}{z}}{x} \]

       2. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right)} \cdot y}{z}}{x} \]

       3. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot y}{z}}{x} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), 1\right) \cdot y}{z}}{x} \]

       5. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, 1\right) \cdot y}{z}}{x} \]

       6. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right)}, 1\right) \cdot y}{z}}{x} \]

       7. unpow2 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       8. *-lowering-*.f64 N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       9. +-commutative N/A

          \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       10. *-commutative N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \frac{1}{720}} + \frac{1}{24}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       11. accelerator-lowering-fma.f64 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{720}, \frac{1}{24}\right)}, \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       12. unpow2 N/A

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{720}, \frac{1}{24}\right), \frac{1}{2}\right), 1\right) \cdot y}{z}}{x} \]

       13. *-lowering-*.f64 100.0

           \[\leadsto \frac{\frac{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right) \cdot y}{z}}{x} \]

   12. Simplified 100.0%

       \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)} \cdot y}{z}}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00028:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)}{x}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+48}:\\ \;\;\;\;y \cdot \frac{\cosh x}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right), 0.5\right), 1\right)}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 70.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \left(x \cdot 0.041666666666666664\right), 1\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= x 2.35e-39)
    (/ (/ y z_m) x)
    (if (<= x 8.2e+101)
      (/
       (* (/ y x) (fma (fma x x 0.0) (* x (* x 0.041666666666666664)) 1.0))
       z_m)
      (* y (/ (* 0.041666666666666664 (* x (* x x))) z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 2.35e-39) {
		tmp = (y / z_m) / x;
	} else if (x <= 8.2e+101) {
		tmp = ((y / x) * fma(fma(x, x, 0.0), (x * (x * 0.041666666666666664)), 1.0)) / z_m;
	} else {
		tmp = y * ((0.041666666666666664 * (x * (x * x))) / z_m);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 2.35e-39)
		tmp = Float64(Float64(y / z_m) / x);
	elseif (x <= 8.2e+101)
		tmp = Float64(Float64(Float64(y / x) * fma(fma(x, x, 0.0), Float64(x * Float64(x * 0.041666666666666664)), 1.0)) / z_m);
	else
		tmp = Float64(y * Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * x))) / z_m));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 2.35e-39], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 8.2e+101], N[(N[(N[(y / x), $MachinePrecision] * N[(N[(x * x + 0.0), $MachinePrecision] * N[(x * N[(x * 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y * N[(N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if x < 2.3500000000000001e-39

   1. Initial program 85.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 63.4

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 63.4%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      5. /-lowering-/.f64 71.1

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]

   7. Applied egg-rr 71.1%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 2.3500000000000001e-39 < x < 8.1999999999999999e101

   1. Initial program 95.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 68.7

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 68.7%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2}}, 1\right) \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

      1. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \frac{1}{24} \cdot \color{blue}{\left(x \cdot x\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(\frac{1}{24} \cdot x\right) \cdot x}, 1\right) \cdot \frac{y}{x}}{z} \]

      3. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(\frac{1}{24} \cdot x\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(\frac{1}{24} \cdot x\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \color{blue}{\left(x \cdot \frac{1}{24}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. *-lowering-*.f64 68.2

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \color{blue}{\left(x \cdot 0.041666666666666664\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

   8. Simplified 68.2%

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot 0.041666666666666664\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

   if 8.1999999999999999e101 < x

   1. Initial program 74.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      2. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \cosh x}}{z \cdot x} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\cosh x}{z \cdot x}} \]

      7. cosh-lowering-cosh.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x}}{z \cdot x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

      9. *-lowering-*.f64 74.5

         \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   4. Applied egg-rr 74.5%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{x \cdot z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x \cdot z} \]

      2. unpow2 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}{x \cdot z} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} + 1}{x \cdot z} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)} + 1}{x \cdot z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x, 1\right)}}{x \cdot z} \]

      6. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}, 1\right)}{x \cdot z} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}, 1\right)}{x \cdot z} \]

      8. +-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}, 1\right)}{x \cdot z} \]

      9. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right), 1\right)}{x \cdot z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x \cdot z} \]

      11. unpow2 N/A

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x \cdot z} \]

      12. *-lowering-*.f64 74.5

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z} \]

   7. Simplified 74.5%

      \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}}{x \cdot z} \]

   8. Taylor expanded in x around inf

      \[\leadsto y \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{3}}{z}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{24} \cdot {x}^{3}}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{24} \cdot {x}^{3}}{z}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{1}{24} \cdot {x}^{3}}}{z} \]

      4. cube-mult N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}}{z} \]

      5. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)}{z} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}}{z} \]

      7. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)}{z} \]

      8. *-lowering-*.f64 100.0

         \[\leadsto y \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)}{z} \]

   10. Simplified 100.0%

       \[\leadsto y \cdot \color{blue}{\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.35 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{+101}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), x \cdot \left(x \cdot 0.041666666666666664\right), 1\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ \begin{array}{l} t_0 := 0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \left(x \cdot t\_0\right)}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t\_0}{z\_m}\\ \end{array} \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (let* ((t_0 (* 0.041666666666666664 (* x (* x x)))))
   (*
    z_s
    (if (<= x 0.00031)
      (/ (/ y z_m) x)
      (if (<= x 5.5e+99) (/ (* (/ y x) (* x t_0)) z_m) (* y (/ t_0 z_m)))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double t_0 = 0.041666666666666664 * (x * (x * x));
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else if (x <= 5.5e+99) {
		tmp = ((y / x) * (x * t_0)) / z_m;
	} else {
		tmp = y * (t_0 / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.041666666666666664d0 * (x * (x * x))
    if (x <= 0.00031d0) then
        tmp = (y / z_m) / x
    else if (x <= 5.5d+99) then
        tmp = ((y / x) * (x * t_0)) / z_m
    else
        tmp = y * (t_0 / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double t_0 = 0.041666666666666664 * (x * (x * x));
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else if (x <= 5.5e+99) {
		tmp = ((y / x) * (x * t_0)) / z_m;
	} else {
		tmp = y * (t_0 / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	t_0 = 0.041666666666666664 * (x * (x * x))
	tmp = 0
	if x <= 0.00031:
		tmp = (y / z_m) / x
	elif x <= 5.5e+99:
		tmp = ((y / x) * (x * t_0)) / z_m
	else:
		tmp = y * (t_0 / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	t_0 = Float64(0.041666666666666664 * Float64(x * Float64(x * x)))
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(Float64(y / z_m) / x);
	elseif (x <= 5.5e+99)
		tmp = Float64(Float64(Float64(y / x) * Float64(x * t_0)) / z_m);
	else
		tmp = Float64(y * Float64(t_0 / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	t_0 = 0.041666666666666664 * (x * (x * x));
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = (y / z_m) / x;
	elseif (x <= 5.5e+99)
		tmp = ((y / x) * (x * t_0)) / z_m;
	else
		tmp = y * (t_0 / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := Block[{t$95$0 = N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(z$95$s * If[LessEqual[x, 0.00031], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], If[LessEqual[x, 5.5e+99], N[(N[(N[(y / x), $MachinePrecision] * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y * N[(t$95$0 / z$95$m), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if x < 3.1e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      5. /-lowering-/.f64 72.2

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 3.1e-4 < x < 5.5000000000000002e99

   1. Initial program 92.9%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 51.3

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 51.3%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{4}\right)} \cdot \frac{y}{x}}{z} \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot {x}^{\color{blue}{\left(2 \cdot 2\right)}}\right) \cdot \frac{y}{x}}{z} \]

      2. pow-sqr N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot {x}^{2}\right)}\right) \cdot \frac{y}{x}}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot \frac{y}{x}}{z} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot x\right) \cdot x\right)} \cdot \frac{y}{x}}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot x\right)\right)} \cdot x\right) \cdot \frac{y}{x}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right)\right) \cdot x\right) \cdot \frac{y}{x}}{z} \]

      8. unpow3 N/A

         \[\leadsto \frac{\left(\left(\frac{1}{24} \cdot \color{blue}{{x}^{3}}\right) \cdot x\right) \cdot \frac{y}{x}}{z} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)\right)} \cdot \frac{y}{x}}{z} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(x \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)\right)} \cdot \frac{y}{x}}{z} \]

      11. *-lowering-*.f64 N/A

          \[\leadsto \frac{\left(x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}\right) \cdot \frac{y}{x}}{z} \]

      12. cube-mult N/A

          \[\leadsto \frac{\left(x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)\right) \cdot \frac{y}{x}}{z} \]

      13. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)\right) \cdot \frac{y}{x}}{z} \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \frac{\left(x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)\right) \cdot \frac{y}{x}}{z} \]

      15. unpow2 N/A

          \[\leadsto \frac{\left(x \cdot \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \cdot \frac{y}{x}}{z} \]

      16. *-lowering-*.f64 51.3

          \[\leadsto \frac{\left(x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)\right) \cdot \frac{y}{x}}{z} \]

   8. Simplified 51.3%

      \[\leadsto \frac{\color{blue}{\left(x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)} \cdot \frac{y}{x}}{z} \]

   if 5.5000000000000002e99 < x

   1. Initial program 74.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      2. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \cosh x}}{z \cdot x} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\cosh x}{z \cdot x}} \]

      7. cosh-lowering-cosh.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x}}{z \cdot x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

      9. *-lowering-*.f64 74.5

         \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   4. Applied egg-rr 74.5%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{x \cdot z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x \cdot z} \]

      2. unpow2 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}{x \cdot z} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} + 1}{x \cdot z} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)} + 1}{x \cdot z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x, 1\right)}}{x \cdot z} \]

      6. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}, 1\right)}{x \cdot z} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}, 1\right)}{x \cdot z} \]

      8. +-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}, 1\right)}{x \cdot z} \]

      9. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right), 1\right)}{x \cdot z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x \cdot z} \]

      11. unpow2 N/A

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x \cdot z} \]

      12. *-lowering-*.f64 74.5

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z} \]

   7. Simplified 74.5%

      \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}}{x \cdot z} \]

   8. Taylor expanded in x around inf

      \[\leadsto y \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{3}}{z}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{24} \cdot {x}^{3}}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{24} \cdot {x}^{3}}{z}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{1}{24} \cdot {x}^{3}}}{z} \]

      4. cube-mult N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}}{z} \]

      5. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)}{z} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}}{z} \]

      7. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)}{z} \]

      8. *-lowering-*.f64 100.0

         \[\leadsto y \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)}{z} \]

   10. Simplified 100.0%

       \[\leadsto y \cdot \color{blue}{\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{y}{x} \cdot \left(x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 89.5% accurate, 2.2× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 6.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z\_m} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 6.7e-19)
    (/
     (/ (* y (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0)) x)
     z_m)
    (/
     (*
      (/ y z_m)
      (fma x (* x (fma x (fma x 0.041666666666666664 0.0) 0.5)) 1.0))
     x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 6.7e-19) {
		tmp = ((y * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0)) / x) / z_m;
	} else {
		tmp = ((y / z_m) * fma(x, (x * fma(x, fma(x, 0.041666666666666664, 0.0), 0.5)), 1.0)) / x;
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 6.7e-19)
		tmp = Float64(Float64(Float64(y * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0)) / x) / z_m);
	else
		tmp = Float64(Float64(Float64(y / z_m) * fma(x, Float64(x * fma(x, fma(x, 0.041666666666666664, 0.0), 0.5)), 1.0)) / x);
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 6.7e-19], N[(N[(N[(y * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(N[(y / z$95$m), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 0.041666666666666664 + 0.0), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 6.69999999999999998e-19

   1. Initial program 80.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 72.9

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 72.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot 1} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x}}{z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y} + \frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x}}{z} \]

      4. distribute-rgt-out N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{\left(y \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right)} \cdot {x}^{2}}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot {x}^{2}}{x}}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)}}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + y \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{x}}{z} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{x}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}{x}}}{z} \]

   8. Simplified 86.8%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right) \cdot y}{x}}}{z} \]

   if 6.69999999999999998e-19 < y

   1. Initial program 93.2%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 89.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 89.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{\frac{y}{x}}{z}} \]

      2. associate-/r* N/A

         \[\leadsto \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \color{blue}{\frac{y}{x \cdot z}} \]

      3. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{\color{blue}{x \cdot z + 0}} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{x \cdot z + 0}} \]

   7. Applied egg-rr 87.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right) \cdot \frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   8. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{\color{blue}{x \cdot z}} \]

      2. associate-/l/ N/A

         \[\leadsto \left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \color{blue}{\frac{\frac{y}{z}}{x}} \]

      3. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(\left(x \cdot x + 0\right) \cdot \left(x \cdot \left(x \cdot \frac{1}{24} + 0\right) + \frac{1}{2}\right) + 1\right) \cdot \frac{y}{z}}{x}} \]

   9. Applied egg-rr 95.8%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right) \cdot \frac{y}{z}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.7 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{z} \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 88.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \frac{1}{z\_m}}{x}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= y 4e+99)
    (/
     (/ (* y (fma x (* x (fma (* x x) 0.041666666666666664 0.5)) 1.0)) x)
     z_m)
    (* y (/ (* (fma x (* x 0.5) 1.0) (/ 1.0 z_m)) x)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (y <= 4e+99) {
		tmp = ((y * fma(x, (x * fma((x * x), 0.041666666666666664, 0.5)), 1.0)) / x) / z_m;
	} else {
		tmp = y * ((fma(x, (x * 0.5), 1.0) * (1.0 / z_m)) / x);
	}
	return z_s * tmp;
}

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (y <= 4e+99)
		tmp = Float64(Float64(Float64(y * fma(x, Float64(x * fma(Float64(x * x), 0.041666666666666664, 0.5)), 1.0)) / x) / z_m);
	else
		tmp = Float64(y * Float64(Float64(fma(x, Float64(x * 0.5), 1.0) * Float64(1.0 / z_m)) / x));
	end
	return Float64(z_s * tmp)
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[y, 4e+99], N[(N[(N[(y * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + 0.5), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(y * N[(N[(N[(x * N[(x * 0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(1.0 / z$95$m), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < 3.9999999999999999e99

   1. Initial program 83.0%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 75.1

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 75.1%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}}{z} \]
   7. Step-by-step derivation

      1. *-rgt-identity N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot 1} + {x}^{2} \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right)}{x}}{z} \]

      2. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right) + \frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x}}{z} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2}\right) \cdot y} + \frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x}}{z} \]

      4. distribute-rgt-out N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{\left(y \cdot \left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)\right)} \cdot {x}^{2}}{x}}{z} \]

      5. +-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \left(y \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}\right) \cdot {x}^{2}}{x}}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\frac{y \cdot 1 + \color{blue}{y \cdot \left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}\right)}}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\frac{y \cdot 1 + y \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{x}}{z} \]

      8. distribute-lft-in N/A

         \[\leadsto \frac{\frac{\color{blue}{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}}{x}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{y \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)}{x}}}{z} \]

   8. Simplified 87.5%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right) \cdot y}{x}}}{z} \]

   if 3.9999999999999999e99 < y

   1. Initial program 89.5%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{y}{x} \cdot \cosh x}}{z} \]

      2. div-inv N/A

         \[\leadsto \frac{\color{blue}{\left(y \cdot \frac{1}{x}\right)} \cdot \cosh x}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{x} \cdot \cosh x\right)}}{z} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{x} \cdot \cosh x}{z}} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x \cdot \frac{1}{x}}}{z} \]

      8. div-inv N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      9. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{\cosh x}{x}}}{z} \]

      10. cosh-lowering-cosh.f64 99.9

          \[\leadsto y \cdot \frac{\frac{\color{blue}{\cosh x}}{x}}{z} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{y \cdot \frac{\frac{\cosh x}{x}}{z}} \]

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2}}{z} + \frac{1}{z}}{x}} \]
   6. Step-by-step derivation

      1. *-lft-identity N/A

         \[\leadsto y \cdot \frac{\frac{1}{2} \cdot \frac{\color{blue}{1 \cdot {x}^{2}}}{z} + \frac{1}{z}}{x} \]

      2. associate-*l/ N/A

         \[\leadsto y \cdot \frac{\frac{1}{2} \cdot \color{blue}{\left(\frac{1}{z} \cdot {x}^{2}\right)} + \frac{1}{z}}{x} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot \frac{1}{z}\right) \cdot {x}^{2}} + \frac{1}{z}}{x} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right)} + \frac{1}{z}}{x} \]

      5. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{{x}^{2} \cdot \left(\frac{1}{2} \cdot \frac{1}{z}\right) + \frac{1}{z}}{x}} \]

      6. associate-*r* N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left({x}^{2} \cdot \frac{1}{2}\right) \cdot \frac{1}{z}} + \frac{1}{z}}{x} \]

      7. *-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{1}{z} + \frac{1}{z}}{x} \]

      8. distribute-lft1-in N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{1}{z}}}{x} \]

      9. +-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{1}{z}}{x} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{1}{z}}}{x} \]

      11. +-commutative N/A

          \[\leadsto y \cdot \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot \frac{1}{z}}{x} \]

      12. unpow2 N/A

          \[\leadsto y \cdot \frac{\left(\frac{1}{2} \cdot \color{blue}{\left(x \cdot x\right)} + 1\right) \cdot \frac{1}{z}}{x} \]

      13. associate-*r* N/A

          \[\leadsto y \cdot \frac{\left(\color{blue}{\left(\frac{1}{2} \cdot x\right) \cdot x} + 1\right) \cdot \frac{1}{z}}{x} \]

      14. *-commutative N/A

          \[\leadsto y \cdot \frac{\left(\color{blue}{x \cdot \left(\frac{1}{2} \cdot x\right)} + 1\right) \cdot \frac{1}{z}}{x} \]

      15. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \frac{1}{2} \cdot x, 1\right)} \cdot \frac{1}{z}}{x} \]

      16. *-commutative N/A

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 1\right) \cdot \frac{1}{z}}{x} \]

      17. *-lowering-*.f64 N/A

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{2}}, 1\right) \cdot \frac{1}{z}}{x} \]

      18. /-lowering-/.f64 91.5

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \color{blue}{\frac{1}{z}}}{x} \]

   7. Simplified 91.5%

      \[\leadsto y \cdot \color{blue}{\frac{\mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \frac{1}{z}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4 \cdot 10^{+99}:\\ \;\;\;\;\frac{\frac{y \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{\mathsf{fma}\left(x, x \cdot 0.5, 1\right) \cdot \frac{1}{z}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.0% accurate, 3.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= x 0.00031)
    (/ (/ y z_m) x)
    (/ (* y (* 0.041666666666666664 (* x (* x x)))) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else {
		tmp = (y * (0.041666666666666664 * (x * (x * x)))) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 0.00031d0) then
        tmp = (y / z_m) / x
    else
        tmp = (y * (0.041666666666666664d0 * (x * (x * x)))) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else {
		tmp = (y * (0.041666666666666664 * (x * (x * x)))) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 0.00031:
		tmp = (y / z_m) / x
	else:
		tmp = (y * (0.041666666666666664 * (x * (x * x)))) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(Float64(y * Float64(0.041666666666666664 * Float64(x * Float64(x * x)))) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = (y / z_m) / x;
	else
		tmp = (y * (0.041666666666666664 * (x * (x * x)))) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 0.00031], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(N[(y * N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      5. /-lowering-/.f64 72.2

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 3.1e-4 < x

   1. Initial program 78.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 69.2

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   7. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\frac{1}{24} \cdot \color{blue}{\left(y \cdot {x}^{3}\right)}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot y\right) \cdot {x}^{3}}}{z} \]

      3. unpow3 N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot y\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot y\right) \cdot \left(\color{blue}{{x}^{2}} \cdot x\right)}{z} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\frac{1}{24} \cdot y\right) \cdot {x}^{2}\right) \cdot x}}{z} \]

      6. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot \left(y \cdot {x}^{2}\right)\right)} \cdot x}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(\frac{1}{24} \cdot \color{blue}{\left({x}^{2} \cdot y\right)}\right) \cdot x}{z} \]

      8. *-commutative N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \color{blue}{\left(\frac{1}{24} \cdot \left({x}^{2} \cdot y\right)\right)}}{z} \]

      11. unpow2 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)\right)}{z} \]

      12. associate-*l* N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot y\right)\right)}\right)}{z} \]

      13. *-lowering-*.f64 N/A

          \[\leadsto \frac{x \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot y\right)\right)}\right)}{z} \]

      14. *-lowering-*.f64 82.6

          \[\leadsto \frac{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot y\right)}\right)\right)}{z} \]

   8. Simplified 82.6%

      \[\leadsto \frac{\color{blue}{x \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot y\right)\right)\right)}}{z} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(x \cdot \left(x \cdot y\right)\right) \cdot \frac{1}{24}\right)}}{z} \]

      2. associate-*r* N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(\left(x \cdot x\right) \cdot y\right)} \cdot \frac{1}{24}\right)}{z} \]

      3. associate-*l* N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(y \cdot \frac{1}{24}\right)\right)}}{z} \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot \left(y \cdot \frac{1}{24}\right)\right)}}{z} \]

      5. +-rgt-identity N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(x \cdot x + 0\right)} \cdot \left(y \cdot \frac{1}{24}\right)\right)}{z} \]

      6. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)} \cdot \left(y \cdot \frac{1}{24}\right)\right)}{z} \]

      7. *-lowering-*.f64 82.6

         \[\leadsto \frac{x \cdot \left(\mathsf{fma}\left(x, x, 0\right) \cdot \color{blue}{\left(y \cdot 0.041666666666666664\right)}\right)}{z} \]

   10. Applied egg-rr 82.6%

       \[\leadsto \frac{x \cdot \color{blue}{\left(\mathsf{fma}\left(x, x, 0\right) \cdot \left(y \cdot 0.041666666666666664\right)\right)}}{z} \]

   11. Taylor expanded in x around 0

       \[\leadsto \frac{\color{blue}{\frac{1}{24} \cdot \left({x}^{3} \cdot y\right)}}{z} \]
   12. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \frac{\color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right) \cdot y}}{z} \]

       2. *-commutative N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \frac{\color{blue}{y \cdot \left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \frac{y \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{3}\right)}}{z} \]

       5. cube-mult N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}\right)}{z} \]

       6. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)\right)}{z} \]

       7. *-lowering-*.f64 N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}\right)}{z} \]

       8. unpow2 N/A

          \[\leadsto \frac{y \cdot \left(\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{z} \]

       9. *-lowering-*.f64 85.7

          \[\leadsto \frac{y \cdot \left(0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)}{z} \]

   13. Simplified 85.7%

       \[\leadsto \frac{\color{blue}{y \cdot \left(0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)}}{z} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.8% accurate, 3.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= x 0.00031)
    (/ (/ y z_m) x)
    (* y (/ (* 0.041666666666666664 (* x (* x x))) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y * ((0.041666666666666664 * (x * (x * x))) / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 0.00031d0) then
        tmp = (y / z_m) / x
    else
        tmp = y * ((0.041666666666666664d0 * (x * (x * x))) / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else {
		tmp = y * ((0.041666666666666664 * (x * (x * x))) / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 0.00031:
		tmp = (y / z_m) / x
	else:
		tmp = y * ((0.041666666666666664 * (x * (x * x))) / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(y * Float64(Float64(0.041666666666666664 * Float64(x * Float64(x * x))) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = (y / z_m) / x;
	else
		tmp = y * ((0.041666666666666664 * (x * (x * x))) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 0.00031], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(y * N[(N[(0.041666666666666664 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      5. /-lowering-/.f64 72.2

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 3.1e-4 < x

   1. Initial program 78.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\cosh x \cdot y}{x}}}{z} \]

      2. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\cosh x \cdot y}{z \cdot x}} \]

      3. *-commutative N/A

         \[\leadsto \frac{\color{blue}{y \cdot \cosh x}}{z \cdot x} \]

      4. associate-/l* N/A

         \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{z \cdot x}} \]

      6. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\cosh x}{z \cdot x}} \]

      7. cosh-lowering-cosh.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\cosh x}}{z \cdot x} \]

      8. *-commutative N/A

         \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

      9. *-lowering-*.f64 73.8

         \[\leadsto y \cdot \frac{\cosh x}{\color{blue}{x \cdot z}} \]

   4. Applied egg-rr 73.8%

      \[\leadsto \color{blue}{y \cdot \frac{\cosh x}{x \cdot z}} \]

   5. Taylor expanded in x around 0

      \[\leadsto y \cdot \frac{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}}{x \cdot z} \]
   6. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto y \cdot \frac{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}}{x \cdot z} \]

      2. unpow2 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}{x \cdot z} \]

      3. associate-*l* N/A

         \[\leadsto y \cdot \frac{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} + 1}{x \cdot z} \]

      4. *-commutative N/A

         \[\leadsto y \cdot \frac{x \cdot \color{blue}{\left(\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x\right)} + 1}{x \cdot z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot x, 1\right)}}{x \cdot z} \]

      6. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}, 1\right)}{x \cdot z} \]

      7. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}, 1\right)}{x \cdot z} \]

      8. +-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}\right)}, 1\right)}{x \cdot z} \]

      9. *-commutative N/A

         \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}\right), 1\right)}{x \cdot z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{24}, \frac{1}{2}\right)}, 1\right)}{x \cdot z} \]

      11. unpow2 N/A

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{24}, \frac{1}{2}\right), 1\right)}{x \cdot z} \]

      12. *-lowering-*.f64 69.1

          \[\leadsto y \cdot \frac{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 0.041666666666666664, 0.5\right), 1\right)}{x \cdot z} \]

   7. Simplified 69.1%

      \[\leadsto y \cdot \frac{\color{blue}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.041666666666666664, 0.5\right), 1\right)}}{x \cdot z} \]

   8. Taylor expanded in x around inf

      \[\leadsto y \cdot \color{blue}{\left(\frac{1}{24} \cdot \frac{{x}^{3}}{z}\right)} \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{24} \cdot {x}^{3}}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto y \cdot \color{blue}{\frac{\frac{1}{24} \cdot {x}^{3}}{z}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\color{blue}{\frac{1}{24} \cdot {x}^{3}}}{z} \]

      4. cube-mult N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}}{z} \]

      5. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{{x}^{2}}\right)}{z} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \color{blue}{\left(x \cdot {x}^{2}\right)}}{z} \]

      7. unpow2 N/A

         \[\leadsto y \cdot \frac{\frac{1}{24} \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)}{z} \]

      8. *-lowering-*.f64 85.7

         \[\leadsto y \cdot \frac{0.041666666666666664 \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)}{z} \]

   10. Simplified 85.7%

       \[\leadsto y \cdot \color{blue}{\frac{0.041666666666666664 \cdot \left(x \cdot \left(x \cdot x\right)\right)}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 68.6% accurate, 3.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;0.041666666666666664 \cdot \frac{x \cdot \left(x \cdot \left(x \cdot y\right)\right)}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (*
  z_s
  (if (<= x 0.00031)
    (/ (/ y z_m) x)
    (* 0.041666666666666664 (/ (* x (* x (* x y))) z_m)))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else {
		tmp = 0.041666666666666664 * ((x * (x * (x * y))) / z_m);
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 0.00031d0) then
        tmp = (y / z_m) / x
    else
        tmp = 0.041666666666666664d0 * ((x * (x * (x * y))) / z_m)
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else {
		tmp = 0.041666666666666664 * ((x * (x * (x * y))) / z_m);
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 0.00031:
		tmp = (y / z_m) / x
	else:
		tmp = 0.041666666666666664 * ((x * (x * (x * y))) / z_m)
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(0.041666666666666664 * Float64(Float64(x * Float64(x * Float64(x * y))) / z_m));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = (y / z_m) / x;
	else
		tmp = 0.041666666666666664 * ((x * (x * (x * y))) / z_m);
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 0.00031], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(0.041666666666666664 * N[(N[(x * N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      5. /-lowering-/.f64 72.2

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 3.1e-4 < x

   1. Initial program 78.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)} \cdot \frac{y}{x}}{z} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1\right)} \cdot \frac{y}{x}}{z} \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right)} \cdot \frac{y}{x}}{z} \]

      3. +-rgt-identity N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} + 0}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x} + 0, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, x, 0\right)}, \frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      6. +-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, 1\right) \cdot \frac{y}{x}}{z} \]

      7. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{{x}^{2} \cdot \frac{1}{24}} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      8. unpow2 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{24} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      9. associate-*l* N/A

         \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{x \cdot \left(x \cdot \frac{1}{24}\right)} + \frac{1}{2}, 1\right) \cdot \frac{y}{x}}{z} \]

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{24}, \frac{1}{2}\right)}, 1\right) \cdot \frac{y}{x}}{z} \]

      11. +-rgt-identity N/A

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{24} + 0}, \frac{1}{2}\right), 1\right) \cdot \frac{y}{x}}{z} \]

      12. accelerator-lowering-fma.f64 69.2

          \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.041666666666666664, 0\right)}, 0.5\right), 1\right) \cdot \frac{y}{x}}{z} \]

   5. Simplified 69.2%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x, 0\right), \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.041666666666666664, 0\right), 0.5\right), 1\right)} \cdot \frac{y}{x}}{z} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{24} \cdot \frac{{x}^{3} \cdot y}{z}} \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \frac{1}{24} \cdot \color{blue}{\frac{{x}^{3} \cdot y}{z}} \]

      3. cube-mult N/A

         \[\leadsto \frac{1}{24} \cdot \frac{\color{blue}{\left(x \cdot \left(x \cdot x\right)\right)} \cdot y}{z} \]

      4. unpow2 N/A

         \[\leadsto \frac{1}{24} \cdot \frac{\left(x \cdot \color{blue}{{x}^{2}}\right) \cdot y}{z} \]

      5. associate-*l* N/A

         \[\leadsto \frac{1}{24} \cdot \frac{\color{blue}{x \cdot \left({x}^{2} \cdot y\right)}}{z} \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{24} \cdot \frac{\color{blue}{x \cdot \left({x}^{2} \cdot y\right)}}{z} \]

      7. unpow2 N/A

         \[\leadsto \frac{1}{24} \cdot \frac{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot y\right)}{z} \]

      8. associate-*l* N/A

         \[\leadsto \frac{1}{24} \cdot \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot y\right)\right)}}{z} \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{24} \cdot \frac{x \cdot \color{blue}{\left(x \cdot \left(x \cdot y\right)\right)}}{z} \]

      10. *-lowering-*.f64 82.6

          \[\leadsto 0.041666666666666664 \cdot \frac{x \cdot \left(x \cdot \color{blue}{\left(x \cdot y\right)}\right)}{z} \]

   8. Simplified 82.6%

      \[\leadsto \color{blue}{0.041666666666666664 \cdot \frac{x \cdot \left(x \cdot \left(x \cdot y\right)\right)}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 59.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{z\_m}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right)}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 0.00031) (/ (/ y z_m) x) (/ (* 0.5 (* x y)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else {
		tmp = (0.5 * (x * y)) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 0.00031d0) then
        tmp = (y / z_m) / x
    else
        tmp = (0.5d0 * (x * y)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / z_m) / x;
	} else {
		tmp = (0.5 * (x * y)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 0.00031:
		tmp = (y / z_m) / x
	else:
		tmp = (0.5 * (x * y)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(Float64(y / z_m) / x);
	else
		tmp = Float64(Float64(0.5 * Float64(x * y)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = (y / z_m) / x;
	else
		tmp = (0.5 * (x * y)) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 0.00031], N[(N[(y / z$95$m), $MachinePrecision] / x), $MachinePrecision], N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

      5. /-lowering-/.f64 72.2

         \[\leadsto \frac{\color{blue}{\frac{y}{z}}}{x} \]

   7. Applied egg-rr 72.2%

      \[\leadsto \color{blue}{\frac{\frac{y}{z}}{x}} \]

   if 3.1e-4 < x

   1. Initial program 78.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, x, 0\right), 1\right)}{\mathsf{fma}\left(x, z, 0\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]

      4. /-lowering-/.f64 35.8

         \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\frac{y}{z}}\right) \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \frac{y}{z}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z}\right) \cdot \frac{1}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \frac{1}{2}}{z}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \frac{1}{2}}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      6. *-lowering-*.f64 48.1

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot 0.5}{z} \]

   10. Applied egg-rr 48.1%

       \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 66.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{z}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 57.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{x}}{z\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right)}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 0.00031) (/ (/ y x) z_m) (/ (* 0.5 (* x y)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / x) / z_m;
	} else {
		tmp = (0.5 * (x * y)) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 0.00031d0) then
        tmp = (y / x) / z_m
    else
        tmp = (0.5d0 * (x * y)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = (y / x) / z_m;
	} else {
		tmp = (0.5 * (x * y)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 0.00031:
		tmp = (y / x) / z_m
	else:
		tmp = (0.5 * (x * y)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(Float64(y / x) / z_m);
	else
		tmp = Float64(Float64(0.5 * Float64(x * y)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = (y / x) / z_m;
	else
		tmp = (0.5 * (x * y)) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 0.00031], N[(N[(y / x), $MachinePrecision] / z$95$m), $MachinePrecision], N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 64.8

         \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   5. Simplified 64.8%

      \[\leadsto \frac{\color{blue}{\frac{y}{x}}}{z} \]

   if 3.1e-4 < x

   1. Initial program 78.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, x, 0\right), 1\right)}{\mathsf{fma}\left(x, z, 0\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]

      4. /-lowering-/.f64 35.8

         \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\frac{y}{z}}\right) \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \frac{y}{z}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z}\right) \cdot \frac{1}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \frac{1}{2}}{z}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \frac{1}{2}}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      6. *-lowering-*.f64 48.1

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot 0.5}{z} \]

   10. Applied egg-rr 48.1%

       \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{\frac{y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 57.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{y}{z\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right)}{z\_m}\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 0.00031) (/ y (* z_m x)) (/ (* 0.5 (* x y)) z_m))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = y / (z_m * x);
	} else {
		tmp = (0.5 * (x * y)) / z_m;
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 0.00031d0) then
        tmp = y / (z_m * x)
    else
        tmp = (0.5d0 * (x * y)) / z_m
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = y / (z_m * x);
	} else {
		tmp = (0.5 * (x * y)) / z_m;
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 0.00031:
		tmp = y / (z_m * x)
	else:
		tmp = (0.5 * (x * y)) / z_m
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(y / Float64(z_m * x));
	else
		tmp = Float64(Float64(0.5 * Float64(x * y)) / z_m);
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = y / (z_m * x);
	else
		tmp = (0.5 * (x * y)) / z_m;
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 0.00031], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision], N[(N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision] / z$95$m), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. *-lowering-*.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

   7. Applied egg-rr 64.8%

      \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

   if 3.1e-4 < x

   1. Initial program 78.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, x, 0\right), 1\right)}{\mathsf{fma}\left(x, z, 0\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]

      4. /-lowering-/.f64 35.8

         \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\frac{y}{z}}\right) \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \frac{y}{z}\right)} \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(x \cdot \frac{y}{z}\right) \cdot \frac{1}{2}} \]

      2. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{x \cdot y}{z}} \cdot \frac{1}{2} \]

      3. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \frac{1}{2}}{z}} \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot \frac{1}{2}}{z}} \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right) \cdot \frac{1}{2}}}{z} \]

      6. *-lowering-*.f64 48.1

         \[\leadsto \frac{\color{blue}{\left(x \cdot y\right)} \cdot 0.5}{z} \]

   10. Applied egg-rr 48.1%

       \[\leadsto \color{blue}{\frac{\left(x \cdot y\right) \cdot 0.5}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{y}{z \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5 \cdot \left(x \cdot y\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 55.2% accurate, 4.6× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \begin{array}{l} \mathbf{if}\;x \leq 0.00031:\\ \;\;\;\;\frac{y}{z\_m \cdot x}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{y}{z\_m}\right)\\ \end{array} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m)
 :precision binary64
 (* z_s (if (<= x 0.00031) (/ y (* z_m x)) (* 0.5 (* x (/ y z_m))))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = y / (z_m * x);
	} else {
		tmp = 0.5 * (x * (y / z_m));
	}
	return z_s * tmp;
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    real(8) :: tmp
    if (x <= 0.00031d0) then
        tmp = y / (z_m * x)
    else
        tmp = 0.5d0 * (x * (y / z_m))
    end if
    code = z_s * tmp
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	double tmp;
	if (x <= 0.00031) {
		tmp = y / (z_m * x);
	} else {
		tmp = 0.5 * (x * (y / z_m));
	}
	return z_s * tmp;
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	tmp = 0
	if x <= 0.00031:
		tmp = y / (z_m * x)
	else:
		tmp = 0.5 * (x * (y / z_m))
	return z_s * tmp

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	tmp = 0.0
	if (x <= 0.00031)
		tmp = Float64(y / Float64(z_m * x));
	else
		tmp = Float64(0.5 * Float64(x * Float64(y / z_m)));
	end
	return Float64(z_s * tmp)
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp_2 = code(z_s, x, y, z_m)
	tmp = 0.0;
	if (x <= 0.00031)
		tmp = y / (z_m * x);
	else
		tmp = 0.5 * (x * (y / z_m));
	end
	tmp_2 = z_s * tmp;
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * If[LessEqual[x, 0.00031], N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(x * N[(y / z$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.1e-4

   1. Initial program 85.8%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

      2. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

      3. accelerator-lowering-fma.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

   5. Simplified 64.8%

      \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

      2. *-commutative N/A

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

      3. *-lowering-*.f64 64.8

         \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

   7. Applied egg-rr 64.8%

      \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

   if 3.1e-4 < x

   1. Initial program 78.7%

      \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z} + \frac{y}{z}}{x}} \]
   4. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{{x}^{2} \cdot y}{z} \cdot \frac{1}{2}} + \frac{y}{z}}{x} \]

      2. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot \frac{{x}^{2} \cdot y}{z}} + \frac{y}{z}}{x} \]

      3. associate-/l* N/A

         \[\leadsto \frac{\frac{1}{2} \cdot \color{blue}{\left({x}^{2} \cdot \frac{y}{z}\right)} + \frac{y}{z}}{x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2}\right) \cdot \frac{y}{z}} + \frac{y}{z}}{x} \]

      5. distribute-lft1-in N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right) \cdot \frac{y}{z}}}{x} \]

      6. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(1 + \frac{1}{2} \cdot {x}^{2}\right)} \cdot \frac{y}{z}}{x} \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{1 + \frac{1}{2} \cdot {x}^{2}}{x} \cdot \frac{y}{z}} \]

      8. times-frac N/A

         \[\leadsto \color{blue}{\frac{\left(1 + \frac{1}{2} \cdot {x}^{2}\right) \cdot y}{x \cdot z}} \]

      9. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{1}{2} \cdot {x}^{2} + 1\right)} \cdot y}{x \cdot z} \]

      10. distribute-rgt1-in N/A

          \[\leadsto \frac{\color{blue}{y + \left(\frac{1}{2} \cdot {x}^{2}\right) \cdot y}}{x \cdot z} \]

      11. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\frac{1}{2} \cdot \left({x}^{2} \cdot y\right)}}{x \cdot z} \]

      12. *-commutative N/A

          \[\leadsto \frac{y + \frac{1}{2} \cdot \color{blue}{\left(y \cdot {x}^{2}\right)}}{x \cdot z} \]

      13. associate-*r* N/A

          \[\leadsto \frac{y + \color{blue}{\left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}}{x \cdot z} \]

      14. /-lowering-/.f64 N/A

          \[\leadsto \color{blue}{\frac{y + \left(\frac{1}{2} \cdot y\right) \cdot {x}^{2}}{x \cdot z}} \]

   5. Simplified 53.4%

      \[\leadsto \color{blue}{\frac{y \cdot \mathsf{fma}\left(0.5, \mathsf{fma}\left(x, x, 0\right), 1\right)}{\mathsf{fma}\left(x, z, 0\right)}} \]

   6. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]
   7. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\frac{1}{2} \cdot \frac{x \cdot y}{z}} \]

      2. associate-/l* N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(x \cdot \frac{y}{z}\right)} \]

      4. /-lowering-/.f64 35.8

         \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\frac{y}{z}}\right) \]

   8. Simplified 35.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(x \cdot \frac{y}{z}\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 48.8% accurate, 7.5× speedup

Math

\[\begin{array}{l} z\_m = \left|z\right| \\ z\_s = \mathsf{copysign}\left(1, z\right) \\ z\_s \cdot \frac{y}{z\_m \cdot x} \end{array} \]

FPCore

z\_m = (fabs.f64 z)
z\_s = (copysign.f64 #s(literal 1 binary64) z)
(FPCore (z_s x y z_m) :precision binary64 (* z_s (/ y (* z_m x))))

C

z\_m = fabs(z);
z\_s = copysign(1.0, z);
double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (z_m * x));
}

Fortran

z\_m = abs(z)
z\_s = copysign(1.0d0, z)
real(8) function code(z_s, x, y, z_m)
    real(8), intent (in) :: z_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z_m
    code = z_s * (y / (z_m * x))
end function

Java

z\_m = Math.abs(z);
z\_s = Math.copySign(1.0, z);
public static double code(double z_s, double x, double y, double z_m) {
	return z_s * (y / (z_m * x));
}

Python

z\_m = math.fabs(z)
z\_s = math.copysign(1.0, z)
def code(z_s, x, y, z_m):
	return z_s * (y / (z_m * x))

Julia

z\_m = abs(z)
z\_s = copysign(1.0, z)
function code(z_s, x, y, z_m)
	return Float64(z_s * Float64(y / Float64(z_m * x)))
end

MATLAB

z\_m = abs(z);
z\_s = sign(z) * abs(1.0);
function tmp = code(z_s, x, y, z_m)
	tmp = z_s * (y / (z_m * x));
end

Wolfram

z\_m = N[Abs[z], $MachinePrecision]
z\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[z]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[z$95$s_, x_, y_, z$95$m_] := N[(z$95$s * N[(y / N[(z$95$m * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 84.1%

   \[\frac{\cosh x \cdot \frac{y}{x}}{z} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{x \cdot z}} \]

   2. +-rgt-identity N/A

      \[\leadsto \frac{y}{\color{blue}{x \cdot z + 0}} \]

   3. accelerator-lowering-fma.f64 50.7

      \[\leadsto \frac{y}{\color{blue}{\mathsf{fma}\left(x, z, 0\right)}} \]

5. Simplified 50.7%

   \[\leadsto \color{blue}{\frac{y}{\mathsf{fma}\left(x, z, 0\right)}} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

      \[\leadsto \frac{y}{\color{blue}{x \cdot z}} \]

   2. *-commutative N/A

      \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

   3. *-lowering-*.f64 50.7

      \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]

7. Applied egg-rr 50.7%

   \[\leadsto \frac{y}{\color{blue}{z \cdot x}} \]
8. Add Preprocessing

Developer Target 1: 97.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\frac{y}{z}}{x} \cdot \cosh x\\ \mathbf{if}\;y < -4.618902267687042 \cdot 10^{-52}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 1.038530535935153 \cdot 10^{-39}:\\ \;\;\;\;\frac{\frac{\cosh x \cdot y}{x}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (/ y z) x) (cosh x))))
   (if (< y -4.618902267687042e-52)
     t_0
     (if (< y 1.038530535935153e-39) (/ (/ (* (cosh x) y) x) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y / z) / x) * cosh(x)
    if (y < (-4.618902267687042d-52)) then
        tmp = t_0
    else if (y < 1.038530535935153d-39) then
        tmp = ((cosh(x) * y) / x) / z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y / z) / x) * Math.cosh(x);
	double tmp;
	if (y < -4.618902267687042e-52) {
		tmp = t_0;
	} else if (y < 1.038530535935153e-39) {
		tmp = ((Math.cosh(x) * y) / x) / z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y / z) / x) * math.cosh(x)
	tmp = 0
	if y < -4.618902267687042e-52:
		tmp = t_0
	elif y < 1.038530535935153e-39:
		tmp = ((math.cosh(x) * y) / x) / z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y / z) / x) * cosh(x))
	tmp = 0.0
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = Float64(Float64(Float64(cosh(x) * y) / x) / z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y / z) / x) * cosh(x);
	tmp = 0.0;
	if (y < -4.618902267687042e-52)
		tmp = t_0;
	elseif (y < 1.038530535935153e-39)
		tmp = ((cosh(x) * y) / x) / z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y / z), $MachinePrecision] / x), $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision]}, If[Less[y, -4.618902267687042e-52], t$95$0, If[Less[y, 1.038530535935153e-39], N[(N[(N[(N[Cosh[x], $MachinePrecision] * y), $MachinePrecision] / x), $MachinePrecision] / z), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024196 
(FPCore (x y z)
  :name "Linear.Quaternion:$ctan from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -2309451133843521/5000000000000000000000000000000000000000000000000000000000000000000) (* (/ (/ y z) x) (cosh x)) (if (< y 1038530535935153/1000000000000000000000000000000000000000000000000000000) (/ (/ (* (cosh x) y) x) z) (* (/ (/ y z) x) (cosh x)))))

  (/ (* (cosh x) (/ y x)) z))